# An introduction to JPEG compression using MATLAB - PDF

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```					    An introduction to JPEG compression using
MATLAB
Arno Swart
30 October, 2003

1    Introduction
This document describes the popular JPEG still image coding format. The
aim is to compress images while maintaining acceptable image quality. This is
achieved by dividing the image in blocks of 8 × 8 pixels and applying a discrete
cosine transform (DCT) on the partitioned image. The resulting coeﬃcients
are quantised, less signiﬁcant coeﬃcients are set to zero. After quantisation
two encoding steps are made, zero run length encoding (RLE) followed by an
entropy coding. Part of the JPEG encoding is lossy (information may get lost.
Therefore, the reconstruction may not be exact), part is lossless (no loss of
information).
JPEG allows for some ﬂexibility in the diﬀerent stages, not every option
is explored. Also the focus is on the DCT and quantisation. The entropy
coding will only be brieﬂy discussed and the ﬁle structure deﬁnitions will not
be considered.
There is a MATLAB implementation of (a subset of) JPEG that can be used
alongside this document. It can be down-loaded from the course’s internet pages,
follow the link to Wavelet & FFT resources. Try understanding the diﬀerent
functions in the source code while reading the document. Some questions will
ask you to modify the code slightly.

2    Tiling and the DCT
To simplify matters we will assume that the width and height of the image are a
multiple of 8 and that the image is either gray scale or RGB (red, green, blue).
Each pixel is assigned a value between 0 and 255, or a triplet of RGB values.
In the case of a color RGB picture a pointwise transform is made to the
YUV (luminance, blue chrominance, red chrominance) color space. This space
is in some sense more decorrelated than the RGB space and will allow for better

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quantisation later. The transform is given by
                                       
Y         0.299     0.587      0.114   R      0
U  = −0.1687 −0.3313           0.5  G + 0.5 ,                      (1)
V          0.5     −0.4187 −0.0813     B     0.5

and the inverse transform is
                               
R       1     0   1.402     Y
G = 1 −0.34414 −0.71414 U − 0.5 .                                (2)
B        1   1.772   0     V − 0.5

The next step is dividing each component of the image in blocks of 8 × 8 pixels.
This is needed since the DCT has no locality information whatsoever. There
is only frequency information available. We continue by shifting every block by
−128, this makes the values symmetric around zero with average close to zero.
The DCT of a signal f (x, y) and the inverse DCT (IDCT) of coeﬃcients
F (u, v) are given by
7     7                 (2x+1)uπ
F (u, v) =     1
4 C(u)C(v)  x=0   y=0 f (x, y) cos(     16   ) cos( (2y+1)vπ ),
16
7    7                           (2x+1)uπ
f (x, y) =     1
4   u=0  v=0 C(u)C(v)F (u, v) cos(      16    ) cos( (2y+1)vπ ),
16
(3)
where                                   √
1/ 2 if z = 0,
C(z) =                                                    (4)
1    otherwise.

Question 0 What is the relation between YUV and f ?

Question 1 The DCT is ‘half ’ of the Fourier transform. Why not use the full
Fourier representation instead of the cosine basis functions? It is because the
DCT can approximate linear functions with less coeﬃcients. To see this take
the array 8, 16, 24, 32, 40, 48, 56, 64. Do a forward Fourier transform, set the last
4 bits to zero (this is what will happen in the quantisation step explained later)
and do a inverse Fourier transform. Perform the same procedure for the DCT.
What do you notice? Can you explain the results?

Question 2 The DCT is given as a two dimensional transform. For imple-
mentation purposes it is more eﬃcient to write it as a sequence of two one-
dimensional transforms. Derive and implement this.

Question 3 The DCT only uses 64 basis functions. One can precompute these
and reuse them in the algorithm at the expense of a little extra memory. A
function that returns the basis functions is provided with the source. Build this
in to the main code.

The blocks are processed row by row, from left to right.

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3      Quantisation and Zigzag scan
So far there has been no compression. We will now proceed by introducing
zeros in the arrays by means of quantisation. Every coeﬃcient cij is divided
by an integer qij and rounded to the nearest integer. The importance of the
coeﬃcients is dependent on the human visual system, the eye is much more
sensitive to low frequencies. Measurements led to standard quantisation tables,
listed below. Note that the tables for luminance and chrominance are diﬀerent
(for gray scale images only the luminance tables are used). The tables are 1
                                      
16 11 10 16 24          40   51   61
12 12 14 19 26           58   60   55 
                                      
14 13 16 24 40           57   69   56 
                                      
1 14 17 22 29 51           87   80   62 
QLU M =                                                     (5)
s 18 22 37 56 68 109 103 77            
24 35 55 64 81 104 113 92 
                                      
49 64 78 87 103 121 120 101
72 92 95 98 112 100 103 99
                                  
17 18 24 47 99 99 99 99
18 21 26 66 99 99 99 99
                                  
24 26 56 99 99 99 99 99
                                  
1 47 66 99 99 99 99 99 99
QCHR =                                                     (6)
s 99 99 99 99 99 99 99 99
                                  
99 99 99 99 99 99 99 99
                                  
99 99 99 99 99 99 99 99
99 99 99 99 99 99 99 99
(In the actual computations, qij is an integer which value is obtained by rounding
to the nearest integer and taking the maximum with 1. Why maximum with
1?) The variable s in the above formulas stands for scale. The higher s, the
fewer coeﬃcients will be set to zero. This is also known as the quality level.

Question 4 Another more primitive quantisation technique is thresholding. Ev-
ery coeﬃcient is divided by the same constant value (all qij are equal). Imple-
ment this, can you ﬁnd a noticeable diﬀerence?

The next ingredient is the so-called zigzag scan. Every block is converted to
a vector by gathering elements in zigzag fashion. See ﬁgure (3) for the zigzag
scheme.
This method has the advantage that low-frequency coeﬃents are placed ﬁrst,
while high frequency components come last in the array. This is likely to intro-
duce long substrings of zeros at the end of the array.
Question 5 What does ‘low-frequency’ and ‘high-frequency’ look like in an im-
age. Can you see the logic in discarding (quantising to zero) the high-frequency
components?
1 The   JPEG standard also allows for inclusion of custom tables in the ﬁle header

3
(0,0)                                      (7,0)

(0,7)                                              (7,7)

Figure 1: The zigzag scan order.

Question 6 In what sense is the YUV color space more decorrolated than the
RGB space (cf. the 2nd paragrapfh of Section 2).

The ﬁrst element of the resulting array is called the DC component. The
other 63 entries are the AC components. They are treated separately in the
entropy coding process.
Question 7 What is the interpretation of the DC coeﬃcient? (In the signal
processing community DC stands for ‘direct current’ while ‘AC’ stands for ‘al-
ternating current’.)

4    RLE and entropy coding
This section describes the ﬁnal step in the compression. We will start with the
DC component, which is simplest. We expect the DC components of a cluster
of neighbouring blocks to diﬀer only slightly from each other. For this reason
we use diﬀerential coding, that is, we visit the DC coeﬃcients of the blocks in
row-by-row, left to right fashion and record the diﬀerence with the previous
DC’s (except for the ﬁrst which remains unchanged). Then, on the array of
diﬀerences, we proceed with Huﬀman coding; for a brief description of Huﬀman
coding, see the next section. Huﬀman coding is an entropy coder.
For the remaining 63 AC coeﬃcients we use zero RLE. This means we ﬁrst
store the number of zeros preceding a nonzero and then the nonzero. For exam-
ple the values 0250001 become the pairs (1, 2), (0, 5), (3, 1). The value (0, 0) is

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a special value signaling that the remaining coeﬃcients are all zero. Since our
zigzag scheme clusters the zeros at the end of the array we probably have good
compression here. Now we come at a point where there is a diﬀerence between
the JPEG standard and our implementation. We use 8 bits per value in the pair
and do Huﬀman coding on the bit stream. The oﬃcial JPEG standard works
diﬀerently, but will probably not have signiﬁcantly more compression.
The standard proposes to store no number greater than 15 in the ﬁrst com-
ponent of the pair. For example 17 zeros and a one would become (15, 0), (2, 1).
Now only four bits (a nibble) are needed for this number. For the second number
we store the number of bits that the coeﬃcient needs. This also needs a max-
imum of 4 bits and we put all bits in a bit stream and apply Huﬀman coding.
After decoding we know the number of bits per number so we can reconstruct
our zero length and bit length values. For the actual coeﬃcients we also make
one bit stream, only taking the number of bits we need. This stream is also
Huﬀman compressed. When decoding we know this number of bits, it came
from our ﬁrst stream, so we can also retrieve these coeﬃcients.

Question 8 What part of the JPEG encoding is lossy and what part is lossless?
How much compression is achieved with the lossless part of JPEG? When
(for what value of s) do we have lossless compression?

5    Huﬀman coding (optional)
Huﬀman coding is the optimal lossless scheme for compressing a bit stream. It
works by ﬁxing an n and dividing the input bit-stream in sequences of length n.
For simplicity of explanation, assign symbols, say A, B, C, , . . ., to all sequences
of zeros and ones (bit-codes) of length n. Now, a bit-stream that we want to
encode might look like ABAAB AACB. Next, the Huﬀman encoder calculates
how often each symbol occurs. Then, the symbols are assigned new bit-codes,
but now not necessarily of length n. The more often a symbol occurs (as the A in
our example), the smaller the length of the new bit-code will be. The output of
the Huﬀman encoder is a bit stream formed by the new bit-coded symbols. For
decoding, we need to know where the bit-code for one symbol stops and a new
one begins. This is solved by enforcing the unique preﬁx condition: the bit-code
of no symbol is the preﬁx of the bit-code of another symbol. For instance, the
four symbols A, B, C and might be coded as in Table 1: A might be coded as
0 in the output bit-stream, while it might represent 00 if n = 2 (assuming there
are only four symbols A, B, C, and ) in the input bit-steam. The number of bits
to represent the string ABAAB AACB of 10 symbols reduces from 10 · 2 = 20
bits to 5 · 1 + 3 · 2 + 1 · 3 + 1 · 3 = 17 bits. Note that the reduction would
be better if A would occur more often than in the example. JPEG allows for
coding according to the number of occurences or according to predeﬁned tables
included in the header of the jpg-picture. There are standard tables compiled
from counting the number of occurences in a large number of images.

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original code   symbol    ouput code      original bit-stream:
00          A       0                 00010000011100001001
01          B       11              symbolized: ABAAB AACB
10          C       100             huﬀman coded bit-stream:
11                  101               01100111010010011

Table 1: If the original input bit-stream is diveded in sequences of length 2 then
the table gives the coding for the four symbols as obtained with the Huﬀman
coding for, for instance, the string ABAAB AACB. Note that reading the
huﬀman coded bit-stream from left to right leads to exact reconstriction of the
original bit-stream.

6     Experiments
Try out for yourself how well JPEG compresses images. The source code comes
with a few images in MATLAB .mat format. Try to make graphs of the scaling
factor versus the number of bits per pixel. Also try to make visual estimations
of the quality. A more quantitative measure would be the mean squared error of
the reconstructed image with respect to the original. Later on you can compare
the results with the performance of the more advanced JPEG-2000 format (it
uses wavelets for compression).

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